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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.decomposition</span></code>.non_negative_factorization</a></li>
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  <div class="section" id="sklearn-decomposition-non-negative-factorization">
<h1><a class="reference internal" href="../classes.html#module-sklearn.decomposition" title="sklearn.decomposition"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.decomposition</span></code></a>.non_negative_factorization<a class="headerlink" href="#sklearn-decomposition-non-negative-factorization" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="sklearn.decomposition.non_negative_factorization">
<code class="sig-prename descclassname">sklearn.decomposition.</code><code class="sig-name descname">non_negative_factorization</code><span class="sig-paren">(</span><em class="sig-param">X</em>, <em class="sig-param">W=None</em>, <em class="sig-param">H=None</em>, <em class="sig-param">n_components=None</em>, <em class="sig-param">init='warn'</em>, <em class="sig-param">update_H=True</em>, <em class="sig-param">solver='cd'</em>, <em class="sig-param">beta_loss='frobenius'</em>, <em class="sig-param">tol=0.0001</em>, <em class="sig-param">max_iter=200</em>, <em class="sig-param">alpha=0.0</em>, <em class="sig-param">l1_ratio=0.0</em>, <em class="sig-param">regularization=None</em>, <em class="sig-param">random_state=None</em>, <em class="sig-param">verbose=0</em>, <em class="sig-param">shuffle=False</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/decomposition/_nmf.py#L844"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.decomposition.non_negative_factorization" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute Non-negative Matrix Factorization (NMF)</p>
<p>Find two non-negative matrices (W, H) whose product approximates the non-
negative matrix X. This factorization can be used for example for
dimensionality reduction, source separation or topic extraction.</p>
<p>The objective function is:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">0.5</span> <span class="o">*</span> <span class="o">||</span><span class="n">X</span> <span class="o">-</span> <span class="n">WH</span><span class="o">||</span><span class="n">_Fro</span><span class="o">^</span><span class="mi">2</span>
<span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">l1_ratio</span> <span class="o">*</span> <span class="o">||</span><span class="n">vec</span><span class="p">(</span><span class="n">W</span><span class="p">)</span><span class="o">||</span><span class="n">_1</span>
<span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">l1_ratio</span> <span class="o">*</span> <span class="o">||</span><span class="n">vec</span><span class="p">(</span><span class="n">H</span><span class="p">)</span><span class="o">||</span><span class="n">_1</span>
<span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">alpha</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">l1_ratio</span><span class="p">)</span> <span class="o">*</span> <span class="o">||</span><span class="n">W</span><span class="o">||</span><span class="n">_Fro</span><span class="o">^</span><span class="mi">2</span>
<span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">alpha</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">l1_ratio</span><span class="p">)</span> <span class="o">*</span> <span class="o">||</span><span class="n">H</span><span class="o">||</span><span class="n">_Fro</span><span class="o">^</span><span class="mi">2</span>
</pre></div>
</div>
<p>Where:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">||</span><span class="n">A</span><span class="o">||</span><span class="n">_Fro</span><span class="o">^</span><span class="mi">2</span> <span class="o">=</span> \<span class="n">sum_</span><span class="p">{</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">}</span> <span class="n">A_</span><span class="p">{</span><span class="n">ij</span><span class="p">}</span><span class="o">^</span><span class="mi">2</span> <span class="p">(</span><span class="n">Frobenius</span> <span class="n">norm</span><span class="p">)</span>
<span class="o">||</span><span class="n">vec</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">||</span><span class="n">_1</span> <span class="o">=</span> \<span class="n">sum_</span><span class="p">{</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">}</span> <span class="nb">abs</span><span class="p">(</span><span class="n">A_</span><span class="p">{</span><span class="n">ij</span><span class="p">})</span> <span class="p">(</span><span class="n">Elementwise</span> <span class="n">L1</span> <span class="n">norm</span><span class="p">)</span>
</pre></div>
</div>
<p>For multiplicative-update (‘mu’) solver, the Frobenius norm
(0.5 * ||X - WH||_Fro^2) can be changed into another beta-divergence loss,
by changing the beta_loss parameter.</p>
<p>The objective function is minimized with an alternating minimization of W
and H. If H is given and update_H=False, it solves for W only.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>X</strong><span class="classifier">array-like, shape (n_samples, n_features)</span></dt><dd><p>Constant matrix.</p>
</dd>
<dt><strong>W</strong><span class="classifier">array-like, shape (n_samples, n_components)</span></dt><dd><p>If init=’custom’, it is used as initial guess for the solution.</p>
</dd>
<dt><strong>H</strong><span class="classifier">array-like, shape (n_components, n_features)</span></dt><dd><p>If init=’custom’, it is used as initial guess for the solution.
If update_H=False, it is used as a constant, to solve for W only.</p>
</dd>
<dt><strong>n_components</strong><span class="classifier">integer</span></dt><dd><p>Number of components, if n_components is not set all features
are kept.</p>
</dd>
<dt><strong>init</strong><span class="classifier">None | ‘random’ | ‘nndsvd’ | ‘nndsvda’ | ‘nndsvdar’ | ‘custom’</span></dt><dd><p>Method used to initialize the procedure.
Default: ‘random’.</p>
<p>The default value will change from ‘random’ to None in version 0.23
to make it consistent with decomposition.NMF.</p>
<p>Valid options:</p>
<ul class="simple">
<li><p>None: ‘nndsvd’ if n_components &lt; n_features, otherwise ‘random’.</p></li>
<li><dl class="simple">
<dt>‘random’: non-negative random matrices, scaled with:</dt><dd><p>sqrt(X.mean() / n_components)</p>
</dd>
</dl>
</li>
<li><dl class="simple">
<dt>‘nndsvd’: Nonnegative Double Singular Value Decomposition (NNDSVD)</dt><dd><p>initialization (better for sparseness)</p>
</dd>
</dl>
</li>
<li><dl class="simple">
<dt>‘nndsvda’: NNDSVD with zeros filled with the average of X</dt><dd><p>(better when sparsity is not desired)</p>
</dd>
</dl>
</li>
<li><dl class="simple">
<dt>‘nndsvdar’: NNDSVD with zeros filled with small random values</dt><dd><p>(generally faster, less accurate alternative to NNDSVDa
for when sparsity is not desired)</p>
</dd>
</dl>
</li>
<li><p>‘custom’: use custom matrices W and H</p></li>
</ul>
</dd>
<dt><strong>update_H</strong><span class="classifier">boolean, default: True</span></dt><dd><p>Set to True, both W and H will be estimated from initial guesses.
Set to False, only W will be estimated.</p>
</dd>
<dt><strong>solver</strong><span class="classifier">‘cd’ | ‘mu’</span></dt><dd><p>Numerical solver to use:</p>
<ul class="simple">
<li><dl class="simple">
<dt>‘cd’ is a Coordinate Descent solver that uses Fast Hierarchical</dt><dd><p>Alternating Least Squares (Fast HALS).</p>
</dd>
</dl>
</li>
<li><p>‘mu’ is a Multiplicative Update solver.</p></li>
</ul>
<div class="versionadded">
<p><span class="versionmodified added">New in version 0.17: </span>Coordinate Descent solver.</p>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 0.19: </span>Multiplicative Update solver.</p>
</div>
</dd>
<dt><strong>beta_loss</strong><span class="classifier">float or string, default ‘frobenius’</span></dt><dd><p>String must be in {‘frobenius’, ‘kullback-leibler’, ‘itakura-saito’}.
Beta divergence to be minimized, measuring the distance between X
and the dot product WH. Note that values different from ‘frobenius’
(or 2) and ‘kullback-leibler’ (or 1) lead to significantly slower
fits. Note that for beta_loss &lt;= 0 (or ‘itakura-saito’), the input
matrix X cannot contain zeros. Used only in ‘mu’ solver.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 0.19.</span></p>
</div>
</dd>
<dt><strong>tol</strong><span class="classifier">float, default: 1e-4</span></dt><dd><p>Tolerance of the stopping condition.</p>
</dd>
<dt><strong>max_iter</strong><span class="classifier">integer, default: 200</span></dt><dd><p>Maximum number of iterations before timing out.</p>
</dd>
<dt><strong>alpha</strong><span class="classifier">double, default: 0.</span></dt><dd><p>Constant that multiplies the regularization terms.</p>
</dd>
<dt><strong>l1_ratio</strong><span class="classifier">double, default: 0.</span></dt><dd><p>The regularization mixing parameter, with 0 &lt;= l1_ratio &lt;= 1.
For l1_ratio = 0 the penalty is an elementwise L2 penalty
(aka Frobenius Norm).
For l1_ratio = 1 it is an elementwise L1 penalty.
For 0 &lt; l1_ratio &lt; 1, the penalty is a combination of L1 and L2.</p>
</dd>
<dt><strong>regularization</strong><span class="classifier">‘both’ | ‘components’ | ‘transformation’ | None</span></dt><dd><p>Select whether the regularization affects the components (H), the
transformation (W), both or none of them.</p>
</dd>
<dt><strong>random_state</strong><span class="classifier">int, RandomState instance or None, optional, default: None</span></dt><dd><p>If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by <code class="docutils literal notranslate"><span class="pre">np.random</span></code>.</p>
</dd>
<dt><strong>verbose</strong><span class="classifier">integer, default: 0</span></dt><dd><p>The verbosity level.</p>
</dd>
<dt><strong>shuffle</strong><span class="classifier">boolean, default: False</span></dt><dd><p>If true, randomize the order of coordinates in the CD solver.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>W</strong><span class="classifier">array-like, shape (n_samples, n_components)</span></dt><dd><p>Solution to the non-negative least squares problem.</p>
</dd>
<dt><strong>H</strong><span class="classifier">array-like, shape (n_components, n_features)</span></dt><dd><p>Solution to the non-negative least squares problem.</p>
</dd>
<dt><strong>n_iter</strong><span class="classifier">int</span></dt><dd><p>Actual number of iterations.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">References</p>
<p>Cichocki, Andrzej, and P. H. A. N. Anh-Huy. “Fast local algorithms for
large scale nonnegative matrix and tensor factorizations.”
IEICE transactions on fundamentals of electronics, communications and
computer sciences 92.3: 708-721, 2009.</p>
<p>Fevotte, C., &amp; Idier, J. (2011). Algorithms for nonnegative matrix
factorization with the beta-divergence. Neural Computation, 23(9).</p>
<p class="rubric">Examples</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">non_negative_factorization</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">W</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">n_iter</span> <span class="o">=</span> <span class="n">non_negative_factorization</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="gp">... </span><span class="n">init</span><span class="o">=</span><span class="s1">&#39;random&#39;</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>

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